the wronskian of the function e^x and e3x is?

Answers

Answer 1

The Wronskian of the functions e and e(3x) is 2e(4x).

The Wronskian of the functions e^x and e^(3x) is found by computing the determinant of the matrix formed by their derivatives. In this case, the Wronskian is:

W(e^x, e^(3x)) = |(d/dx)e^x   (d/dx)e^(3x)|
                      |e^x         3e^(3x)|

Now, compute the determinant:

W(e^x, e^(3x)) = e^x(3e^(3x)) - e^(3x)e^x
W(e^x, e^(3x)) = 2e^(4x)

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Related Questions

calculate the euclidean distance between the following two points: (5,8,-2) and (-4,5,3) round to two decimal places

Answers

The Euclidean distance between the two points (5,8,-2) and (-4,5,3) is approximately 10.72 (rounded to two decimal places).

Explanation: -

To calculate the Euclidean distance between two points (5,8,-2) and (-4,5,3), you can use the following formula:

Euclidean Distance = √((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)

Here, (x1, y1, z1) = (5, 8, -2) and (x2, y2, z2) = (-4, 5, 3). Now, plug in the values:

Step 1. Calculate the differences between the co-ordinates;
  (x2 - x1) = (-4 - 5) = -9
  (y2 - y1) = (5 - 8) = -3
  (z2 - z1) = (3 - (-2)) = 5

Step 2. Square the differences:
  (-9)^2 = 81
  (-3)^2 = 9
  (5)^2 = 25

Step 3. Add the squared differences:
  81 + 9 + 25 = 115

Step 4. Take the square root:
  √115 ≈ 10.72

So, the Euclidean distance between the two points is approximately 10.72 (rounded to two decimal places).

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The radius of a circle is 14 yards. What is the circle's circumference? 3.14 for pi

Answers

Answer:

87.96

Step-by-step explanation:

Where pi is approximately equal to 3.14 and r is the radius of the circle.

In this case, the radius is 14 yards.

So the circumference is:

2pi14 = 87.96 yards.

How many solutions are there to the equation x1 + x2 + x3 + x4 + X's + x6 = 29, where x; i = 1, 2, 3, 4, 5, 6, is a nonnegative integer such that a) x;> 1 for i = 1, 2, 3, 4, 5, 6? b) x 21,2 2,43 3,44 24, X5 > 5, and x 6? c) x 35? d) x < 8 and .x > 8?

Answers

A. The number of solutions to the equation x1 + x2 + x3 + x4 + x5 + x6 = 29, where xi is a nonnegative integer, is: a) 22C6, b) 29C5, c) 23C5, and d) 6C122C5 - 1 - 6C114C5.

B.

a) Since xi>1, we can subtract 2 from each xi, which gives us the equation x1' + x2' + x3' + x4' + x5' + x6' = 17, where xi' = xi - 2. Then, we can use stars and bars to get the solution as (17 + 6 - 1)C(6 - 1) = 22C5.

b) Since x2>1, x5>5, and x6>0, we can subtract 2 from x2, subtract 6 from x5, and subtract 1 from x6, which gives us the equation x1 + x2' + x3 + x4 + x5' + x6' = 20, where xi' = xi - 2 for i = 2, 5, and xi' = xi - 1 for i = 6. Then, we can use stars and bars to get the solution as (20 + 6 - 1)C(6 - 1) = 29C5.

c) Since x3≥5, we can subtract 5 from x3, which gives us the equation x1 + x2 + x3' + x4 + x5 + x6 = 24, where x3' = x3 - 5. Then, we can use stars and bars to get the solution as (24 + 6 - 1)C(6 - 1) = 23C5.

d) To find the number of solutions where xi<8 for all i and xi>8 for at least one i, we can subtract 1 from x5, which gives us the equation x1 + x2 + x3 + x4 + x5' + x6 = 21, where x5' = x5 - 1. Then, we can use stars and bars to get the total number of solutions where xi<8 for all i as (21 + 6 - 1)C(6 - 1) = 26C5. To find the number of solutions where xi<8 for all i and xi>8 for at least one i, we can subtract 9 from xi for the i that is greater than 8 and subtract 1 from x5, which gives us the equation x1 + x2 + x3 + x4 + x5' + x6' = 12, where x5' = x5 - 1 and x_i' = x_i - 9 for i>8. Then, we can use stars and bars to get the total number of solutions where xi<8 for all i as (12 + 6 - 1)C(6 - 1) = 16C5. Therefore, the number of solutions where xi<8 for all i except xi>8 for at least one i is 26C5 - 16C5 = 6C122C5 - 1 - 6C114C5.

Note: In all cases, we use the stars and bars technique to count the number of nonnegative integer solutions to an equation of the form x1 + x2 + ... + xn = k. The answer

The volume of air in a person's lungs can be modeled with a periodic function. The
graph below represents the volume of air, in ml., in a person's lungs over time t,
measured in seconds.
What is the period and what does it represent in this
context?
1000
you
(2-5, 2900)
(5-5, 1100)
Time (in seconds)
(8.5, 2900)
(11.5, 1100)

Answers

The period of the function represent the given context is (8.5, 2900).

The period of this function is 8.5 seconds, and it represents the time it takes for the person's lungs to fill up with air, then empty out again.

The graph shows that the volume of air in the person's lungs is at its maximum (2900 ml) at the start of each period, and then decreases over time until it reaches its minimum (1100 ml) at the end of each period.

Therefore, the period of the function represent the given context is (8.5, 2900).

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A bridge connecting two cities separated by a lake has a length of 3.961 mi.
Use the table of facts to find the length of the bridge in yards.
Round your answer to the nearest tenth.

Answers

The length of the bridge in yards is 6971.36 yards

Find the length of the bridge in yards.

From the question, we have the following parameters that can be used in our computation:

lake has a length of 3.961 mi.

This means that

Lenght = 3.961 mi.

Use the table of facts to find the length of the bridge in yards, we have

1 mile = 1760 yards

Substitute the known values in the above equation, so, we have the following representation

3.961 * 1 mile = 3.961 * 1760 yards

Evaluate the products

3.961 miles = 6971.36 yards

Recall that

Lenght = 3.961 mi.

So, we have

Lenght = 6971.36 yards

Hence, the length is 6971.36 yards

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the proportion of americans with hypertension is about 33%. the claim of lbp inc is that if you take their vitamin, the chances that you get high blood pressure will go down. what are the hypotheses for this?

Answers

The null hypothesis (H0) for this probability would be that there is no effect of taking the vitamin on the chances of getting high blood pressure.

The alternative hypothesis (Ha) would be that taking the vitamin reduces the chances of getting high blood pressure.

Therefore, the hypotheses for this claim can be written as:

H0: The proportion of Americans with hypertension who take the vitamin is the same as the proportion of Americans with hypertension who do not take the vitamin.

Ha: The proportion of Americans with hypertension who take the vitamin is lower than the proportion of Americans with hypertension who do not take the vitamin.

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Need help asap! thanks!

Answers

Yes the opposite sides of the figure are congruent because:

Both WX and YZ have a length of 3.6

Both XY and WZ have a length of 16.8

How to Use Pythagoras Theorem?

Pythagoras Theorem is the way in which you can find the missing length of a right angled triangle.

The triangle has three sides, the hypotenuse (which is always the longest), Opposite (which doesn't touch the hypotenuse) and the adjacent (which is between the opposite and the hypotenuse).

Pythagoras Theorem is in the form of;

a² + b² = c²

Using Pythagoras theorem, we have:

WX = √(3² + 2²)

WX = √13 = 3.6

YZ = √(3² + 2²)

YZ = √13 = 3.6

Similarly:

XY = √(5² + 16²)

XY = √281

XY = 16.8

WZ = √(5² + 16²)

WZ = √281

WZ = 16.8

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Yes the opposite sides of the figure are congruent because:

Both WX and YZ have a length of 3.6

Both XY and WZ have a length of 16.8

How to Use Pythagoras Theorem?

Pythagoras Theorem is the way in which you can find the missing length of a right angled triangle.

The triangle has three sides, the hypotenuse (which is always the longest), Opposite (which doesn't touch the hypotenuse) and the adjacent (which is between the opposite and the hypotenuse).

Pythagoras Theorem is in the form of;

a² + b² = c²

Using Pythagoras theorem, we have:

WX = √(3² + 2²)

WX = √13 = 3.6

YZ = √(3² + 2²)

YZ = √13 = 3.6

Similarly:

XY = √(5² + 16²)

XY = √281

XY = 16.8

WZ = √(5² + 16²)

WZ = √281

WZ = 16.8

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Let A and wDrmine if w is in Col(A). Is w in Nul(A)? -2 4 Determine if w is in Col(A). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The vector w is in Col(A) because the columns of A span R2 O B. The vector w is in Col(A) because Ax w is a consistent system. One solution is x ° C. The vector w is not in Col(A) because Ax = wis an inconsistent system. One row of the reduced row echelon form of the augmented matrix [A 0] has the form [0 0 b] where b The vector w is not in Col(A) because w is a linear combination of the columns of A. ( D.

Answers

B. The vector w is in Col(A) because Ax = w is a consistent system. One solution is x is the correct option.

To determine if w is in Col(A), we need to check if there exists a solution x such that Ax = w. If such a solution exists, then w is in Col(A).

However, we cannot determine if w is in Nul(A) without knowing more information about matrix A. The nullspace of a matrix is the set of all solutions x to the equation Ax = 0. It is possible for a vector to be in the column space but not in the nullspace, and vice versa.

So, the correct choice is:

B. The vector w is in Col(A) because Ax = w is a consistent system. One solution is x = (-2,4)^T.

We cannot determine if w is in Nul(A) without more information about A.

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Find the arithmetic mean of the sequence 10,___,___,___,-18

Answers

To find the arithmetic mean of the sequence 10, ___, ___, ___, -18, we need to fill in the missing values first.

Since the sequence has five terms, and the first and last terms are given, we can use the formula for the arithmetic mean to find the missing terms:

arithmetic mean = (sum of terms) / (number of terms)

We know that the sum of the terms is equal to:

10 + x + y + z - 18

Simplifying:

x + y + z - 8

Since there are five terms in the sequence, we can set the arithmetic mean equal to:

(arithmetic mean) = (10 + x + y + z - 18) / 5

Simplifying:

(arithmetic mean) = (x + y + z - 8) / 5

We can now substitute the given arithmetic mean into the equation and solve for x + y + z:

-5 = (x + y + z - 8) / 5

Multiplying both sides by 5:

-25 = x + y + z - 8

Simplifying:

x + y + z = -17

Now that we know the sum of the terms, we can find the arithmetic mean by dividing by the number of terms:

(arithmetic mean) = (10 + x + y + z - 18) / 5

(arithmetic mean) = (-17 + 10 + (-18) - 18) / 5

(arithmetic mean) = -43 / 5

(arithmetic mean) = -8.6

Therefore, the arithmetic mean of the sequence 10, ___, ___, ___, -18 is -8.6.

Answer this math question for 15 points

Answers

Answer:

(the length of leg)^2=87^2-60^2=3969

so when take the root :

the length of leg=63ft

Determine whether the set, together with the indicated operations, is a vector space. If it is not, then identify one of the vector space axioms that fails. The set of all 4 x 4 matrices of the form [a c b 0] with the standard operations The set is a vector space. The set is not a vector space because it is not closed under addition. The set is not a vector space because an additive inverse does not exist. The set is not a vector space because it is not closed under scalar multiplication. The set is not a vector space because a scalar identity does not exist.

Answers

The set is not a vector space because it is not closed under addition.

To determine whether the given set is a vector space, we need to check if it satisfies the vector space axioms. Let's consider the closure under addition axiom. Let A = [a1 c1 b1 0] and B = [a2 c2 b2 0] be two 4x4 matrices in the set. Adding A and B, we get:

A + B = [a1+a2 c1+c2 b1+b2 0+0] = [a1+a2 c1+c2 b1+b2 0]

The resulting matrix is not guaranteed to be in the set because the sum of b1 and b2 (b1+b2) might not be equal to the sum of a1 and a2 (a1+a2). Therefore, the set is not closed under addition, and it is not a vector space.

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] a bacteria culture starts with e 10 bacteria and grows at a rate proportional to its size. after 3 hours there are e 16 bacteria. 1. find an expression for the number of bacteria after t hours.

Answers

An expression for the number of bacteria after t hours is [tex]N(t) = e^{(10 + 2t)}[/tex].

To find an expression for the number of bacteria after t hours, given that a bacteria culture starts with e¹⁰ bacteria and grows at a rate proportional to its size, we can use the exponential growth formula:
[tex]N(t) = N(0) * e^{(kt)}[/tex]
where N(t) is the number of bacteria after t hours, N(0) is the initial number of bacteria, k is the growth constant, and t is the time in hours.

First, we are given that after 3 hours there are e¹⁶ bacteria.

Let's use this information to find the growth constant k:
[tex]e^{16} = e^{10} * e^{(3k)} \\e^{6} = e^{(3k)}[/tex]

Now, we can find k by dividing both sides by 3:
6 = 3k
k = 2

Now that we have the growth constant k, we can plug it into the exponential growth formula:
[tex]N(t) = e^{10} * e^{(2t)}[/tex]
which is equivalent to:
[tex]N(t) = e^{(10 + 2t)}[/tex]

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TEXT ANSWER
2x + 5y = -10
-2x + y = 46

SHOW ALL WORK in the way that works best for you

Answers

Answer:

x= - 20

y= 6

Step-by-step explanation:

2x + 5y = -10

-2x + y = 46

if u solve it

it is 6y= 36 so y =6

and substitute y value into the equation:

-2x+ 6 = 46

-2x= 40

so x= -20

Need help with question 14 and
15. Explain your final decision regarding your portfolio

Answers

This prompt is about Investment Decision Making. See the attached completed form and the response below.

Why did you make the choice of the following portfolio?

After carefully considering my investment goals, risk tolerance, and market conditions, I have decided to diversify my portfolio between Equity Funds and Bond Funds.

Specifically, I will be allocating my investments towards a mix of funds which includes S&P 500 Index Fund, Big Cap Value Fund, Undaunted Small Cap Fund, International Index Fund, OhMyGosh Performance Fund, Healthcare Fund, and Emerging Markets Index fund in the Equity category.

In terms of the Bond Funds Category; Intermediate Term Treasury Index Fund, Corporate Bond Index Fund and International Active Bond Fund align with my objectives. Ensuring diversification through this portfolio allocation, I am confident in its ability to assist me in achieving my long-term financial goals.

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1 Tiles are laid according to the pattern indicated below. Three figures were drawn. Draw the next figure in the pattern. ​

Answers

Unfortunately, there is not enough information provided to accurately determine the pattern of the tiles. It is also unclear what the three figures drawn look like, so it is impossible to determine the next figure in the sequence. As for the numerical sequence provided, the outlier may skew the mean but it will not affect the range. However, the presence of an outlier may affect measures of central tendency such as the mean and median, leading to potential inaccuracies when analyzing the data. It would be beneficial to have more information and clarification to properly answer this question.

The graph shows the height of a tire's air valve y, in inches, above or below the center of the tire, for a given number of seconds, x.
What is the diameter of the tire?

Answers

The value of diameter of the tire is, 120° per second.

What is an expression?

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Since we have given a figure.

There is period of 2π in 3 seconds.

That means, it covers 2π degrees in every 3 seconds.

As we need to find the number of degrees the valve move per seconds.

We will use "Unitary method":

In every 3 seconds , Number of degrees would be 360°.

So, in every 1 second, Number of degrees would be

Hence, The valve move 120° per second.

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Pls help me asap!!!

On the axes below, make an appropriate scale and graph exactly one cycle of the trigonometric function y = 7 sin 6x.

Answers

The graph is given in the image below:

How to make the right scale for the trig function

To plot a full cycle of y = 7sin(6x), begin by dividing the period (2π) by six to obtain π/3, which is then used to mark every increment of π/6 along the x-axis.

Additionally, since y ranges from -7 to 7, label the y-axis in increments of either 1 or 2.

Plot the key points at (0,0), (π/12,7), (π/6,0), (π/4,-7), and (π/3,0), and finally connect them smoothly with a curve to complete the plot of one full cycle.

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what is equivalent to 5x - 19

Answers

For x = 3, the expression 5x - 19 is equivalent to -4.

Here, we have an expression, 5x - 19, which is not an equation yet because it doesn't have an equal sign (=). An equation requires that both sides of the equal sign be equivalent.

To make this expression into an equation, we need to set it equal to something. Let's say we want to find an expression that is equivalent to 5x - 19. We can represent this unknown expression as "y". So, our equation will be:

5x - 19 = y

This equation is now saying that the expression 5x - 19 is equivalent to y. We can also say that "y" is a function of "x", meaning that the value of "y" depends on the value of "x".

We can use this equation to find the value of "y" for any given value of "x". If we substitute x = 3, we get:

5(3) - 19 = y

y = 15 - 19

y = -4

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Complete Question:

what is equivalent to 5x - 19 when x = 3?

A normally distributed population has mean 30.5 and standard deviation 3.5. Find the mean of sample mean for sample size of 10. a. 30.5 b. 0.35 c. 3.5 d. 35.0 e. 3.05

Answers

The mean of sample mean for sample size of 10 is e. 3.05.

To find the mean of sample means for a sample size of 10 from a normally distributed population with mean 30.5 and standard deviation 3.5, we use the formula:

mean of sample means = population mean = 30.5

So the answer is not affected by the sample size or standard deviation. The mean of the sample means will always be equal to the population mean.

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The steps for circumscribing a circle about a triangle are shown:

Step 1 Construct the angle bisector of one side of the triangle.
Step 2 Construct the perpendicular bisector of another side of the triangle.
Step 3 Indicate the point of intersection of the bisectors with a point representing the center of the circle.
Step 4 Place the compass on the center, adjust its length to reach any vertex of the triangle, and draw your circumscribed circle.

Which step is incorrect, and how can it be fixed?

a
Step 1, replace "angle bisector" with "perpendicular bisector"

b
Step 3, replace "circle" with "triangle"

c
Step 4, replace "vertex" with "side"

d
All steps are correct

Answers

There is an error in Step 1. The correct answer should be: a) Step 1, replace "angle bisector" with "perpendicular bisector"

What do the terms "inscribed" and "circumscribed" figures mean?

An inscribed figure is, in essence, one shape drawn inside of another. A form drawn outside another shape is referred to as a circumscribed figure. All of a polygon's corners, or vertices, need to touch the circle in order for it to be enclosed inside of it.

The appropriate procedures for drawing a circle around a triangle are as follows:

Step 1: Create the perpendicular bisector of one triangle side .

Step 2: Create the perpendicular bisector of a different triangle side .

Step 3: Mark the place where the bisectors cross a point that represents the centre of the circle.

Step 4: Center the compass and adjust its length to reach any triangle vertex before drawing your circle's circumference.

Hence, option a is correct answer.

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what is in the middle of, 26.27 and 26.89?

Answers

The middle value between 26.27 and 26.89 is 26.58.

To find the middle value between 26.27 and 26.89, you can follow these steps:

1. Add the two numbers together: 26.27 + 26.89 = 53.16
2. Divide the sum by 2 to find the average: 53.16 ÷ 2 = 26.58

The middle value between 26.27 and 26.89 is 26.58.

The middle value of a set of numbers is commonly referred to as the "median". The median is a statistical measure of central tendency that represents the value that separates the lower and upper halves of a dataset.

To find the median of a set of numbers, the numbers must first be arranged in order from lowest to highest (or highest to lowest). If the dataset contains an odd number of values, the median is the middle value.

For example, if we have the set of numbers {1, 3, 5, 7, 9}, the median is 5, which is the value that separates the lower half {1, 3} from the upper half {7, 9}.

If the dataset contains an even number of values, the median is the average of the two middle values. For example, if we have the set of numbers {2, 4, 6, 8}, the median is (4 + 6) / 2 = 5, which is the average of the two middle values that separate the lower half {2, 4} from the upper half {6, 8}.

The median is a useful measure of central tendency because it is not affected by extreme values or outliers in the dataset, unlike the mean, which can be skewed by such values.

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The first four nonzero terms in the power series expansion of the function f(x) = sinx about x = 0 are Select the correct answer. Oa 1-x+x2/2-x3 /3 Ob.1-x2/2+x4/24-x6/720 Odx-x3/6+x/120-x/5040 Oe. 1 +x2 / 2 +x4 / 4 +x" / 6

Answers

The first four nonzero terms in the power series expansion option (D) - [tex]x^{3/6} + x^{5/120} - x^{7/5040[/tex]

What is function?

In mathematics, a function is a unique arrangement of the inputs (also referred to as the domain) and their outputs (sometimes referred to as the co-domain), where each input has exactly one output and the output can be linked to its input.

The power series expansion of the function f(x) = sin(x) about x = 0 is given by:

f(x) = [tex]x - x^{3/3}! + x^{5/5}! - x^{7/7}! + ...[/tex]

To find the first four nonzero terms, we substitute x = 0 into the series and discard all the terms that are zero:

f(0) = 0 - 0 + 0 - 0 + ... = 0

f'(0) = 1 - 0 + 0 - 0 + ... = 1

f''(0) = 0 - 3!/2! + 0 + 0 + ... = -3/2

f'''(0) = 0 + 0 + 5!/3! - 0 + ... = 5/6

Therefore, the first four nonzero terms in the power series expansion of sin(x) about x = 0 are:

sin(x) ≈ [tex]x - x^{3/3}! + x^{5/5}! - x^{7/7}![/tex]

Simplifying this expression, we get:

sin(x) ≈ [tex]x - x^{3/6} + x^{5/120} - x^{7/5040[/tex]

So, the correct answer is option (D) - [tex]x^{3/6} + x^{5/120} - x^{7/5040[/tex].

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nicola, stephanie, kathryn, and william, in that order, take turns at rolling a fair die until one of them throws a 6. what is the probability that william is the first to throw a 6?

Answers

The probability that William is the first to throw a 6 is approximately 125/1296 or roughly 0.096.

How to calculate the probability?

You asked for the probability that William is the first to throw a 6 when Nicola, Stephanie, Kathryn, and William take turns rolling a fair die.

To calculate this probability, we need to consider the following events:
1. William rolls a 6.
2. Nicola, Stephanie, and Kathryn all roll a number other than 6.

The probability of each person rolling a 6 is 1/6, while the probability of not rolling a 6 is 5/6.

So, the probability that William is the first to roll a 6 is:

P(William rolls 6) = P(Nicola rolls not 6) × P(Stephanie rolls not 6) × P(Kathryn rolls not 6) × P(William rolls 6)
= (5/6) × (5/6) × (5/6) × (1/6)

Now, multiply the fractions:

= (5 × 5 × 5 × 1) / (6 × 6 × 6 × 6)
= 125 / 1296

So, the probability that William is the first to throw a 6 is approximately 125/1296, or roughly 0.096 (rounded to 3 decimal places).

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calculate constrained minimum find the points on the curve xy2 = 54 nearest the origin.

Answers

To calculate the constrained minimum, we need to use the method of Lagrange multipliers. We define the Lagrangian function L as L(x,y,λ) = xy^2 - λ(d^2 - x^2 - y^2 - z^2), where d represents the distance between the origin and the nearest point on the curve.

Taking the partial derivatives of L with respect to x, y, and λ, we get:

dL/dx = y^2 + 2λx = 0
dL/dy = 2xy - 2λy = 0
dL/dλ = d^2 - x^2 - y^2 - z^2 = 0

Solving these equations simultaneously, we get:

x = ± 3√6, y = ± √6, and λ = 3/2

Therefore, the points on the curve xy^2 = 54 nearest to the origin are:

(3√6, √6) and (-3√6, -√6)

These points are the constrained minimum because they are the closest points on the curve to the origin.
To find the constrained minimum and the points on the curve xy^2 = 54 nearest to the origin, we can use the method of Lagrange multipliers. Let f(x, y) = x^2 + y^2 be the distance squared from the origin, and g(x, y) = xy^2 - 54 as the constraint.

First, calculate the gradients:
∇f(x, y) = (2x, 2y)
∇g(x, y) = (y^2, 2xy)

Now, set ∇f(x, y) = λ ∇g(x, y):
(2x, 2y) = λ(y^2, 2xy)

This gives us two equations:
1) 2x = λy^2
2) 2y = λ2xy

From equation (2), we can get:
λ = 1/y

Now, substitute λ into equation (1):
2x = (1/y)y^2
2x = y

Using the constraint equation g(x, y) = xy^2 - 54 = 0, we can substitute y = 2x:
2x(2x)^2 = 54
8x^3 = 54
x^3 = 27/4
x = ∛(27/4) = 3/√4 = 3/2

Now we can find y using y = 2x:
y = 2(3/2) = 3

Thus, the point nearest to the origin on the curve xy^2 = 54 is (3/2, 3).

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A strip of width 4 cm is attached to one side of a square to form a rectangle. The area of the rectangle formed is 77c * m ^ 2 then find the length of the side of the square.A strip of width 4 cm is attached to one side of a square to form a rectangle. The area of the rectangle formed is 77c * m ^ 2 then find the length of the side of the square.

Answers

Answer:

x = -2 + sqrt(4 + 77c*m^2) cm.

Step-by-step explanation:

Let's denote the side of the square by "x".

When the strip of width 4 cm is attached to one side of the square, the resulting rectangle has dimensions of (x+4) cm by x cm.

The area of the rectangle is given by:

(x+4) * x = 77c * m^2

Expanding the left-hand side and simplifying, we get:

x^2 + 4x = 77c * m^2

Moving all the terms to one side, we get:

x^2 + 4x - 77c * m^2 = 0

Now we can use the quadratic formula to solve for x:

x = [-4 ± sqrt(4^2 - 41(-77cm^2))] / (21)

x = [-4 ± sqrt(16 + 308cm^2)] / 2

x = [-4 ± 2sqrt(4 + 77cm^2)] / 2

x = -2 ± sqrt(4 + 77c*m^2)

Since the length of a side of a square must be positive, we can discard the negative solution and get:

x = -2 + sqrt(4 + 77c*m^2)

Therefore, the length of the side of the square is x = -2 + sqrt(4 + 77c*m^2) cm.

The length of the side of the square is x = -2 + 2√(1 + 19c) cm.

What is Area of Rectangle?

The area of Rectangle is length times of width.

Let the side of the square be x cm.

When a strip of width 4 cm is attached to one side of the square, the resulting rectangle will have dimensions (x + 4) cm and x cm.

The area of the rectangle is given as 77c m² so we have:

(x + 4)x = 77c

Expanding the left side, we get:

x² + 4x = 77c

Bringing all the terms to one side, we have:

x² + 4x - 77c = 0

Now, we can use the quadratic formula to solve for x:

x = [-4 ± √(4² - 4(1)(-77c))] / 2(1)

x = [-4 ± √(16 + 308c)] / 2

x = [-4 ± √(16(1 + 19c))] / 2

x = [-4 ± 4√(1 + 19c)] / 2

x = -2 ± 2√(1 + 19c)

Since x must be positive, we take the positive root:

x = -2 + 2√(1 + 19c)

Therefore, the length of the side of the square is x = -2 + 2√(1 + 19c) cm.

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Boots and Dora are getting aloo parates for their iftaari party. If they paid $75 for 15 aloo parates, what is the unit rate of one aloo parate

Answers

Answer:

To find the unit rate of one aloo parate, we need to divide the total cost of 15 aloo parates by the number of aloo parates.

The cost per aloo parate can be calculated by dividing $75 by 15 as follows:

Cost per aloo parate = $75 ÷ 15 = $5

Therefore, the unit rate of one aloo parate is $5.

Answer:

$5

Step-by-step explanation:

15 aloo parate = $75

1 aloo parate = 75÷15 = 5

So, 1 aloo parate costs $5

hope it helps! byeeee

Evaluate the iterated integral by converting to polar coordinates.
4
∫ ^√16 − x2 e^−x^2 − y^2 dy dx
0
____

Answers

The final answer will be an approximation.

We have the iterated integral:

∫(0 to √16) ∫(0 to √[tex]16-x^2) e^(-x^2-y^2) dy dx[/tex]

To convert to polar coordinates, we use the substitution x = r cos(θ) and y = r sin(θ), where r is the radial distance and θ is the angle with the positive x-axis.

The limits of integration also need to be changed accordingly. For the inner integral, we have:

0 ≤ y ≤ √[tex](16-x^2)[/tex]

Substituting y = r sin(θ), we get:

0 ≤ r sin(θ) ≤ √(16 - [tex]r^2 cos^2[/tex](θ))

Squaring both sides, we get:

0 ≤ [tex]r^2 sin^2[/tex](θ) ≤ 16 - [tex]r^2 cos^2[/tex](θ)

Rearranging, we get:

[tex]r^2 (cos^2[/tex](θ) + [tex]sin^2[/tex](θ)) ≤ 16

[tex]r^2[/tex] ≤ 16

0 ≤ r ≤ 4

For the outer integral, we have:

0 ≤ x ≤ √16

Substituting x = r cos(θ), we get:

0 ≤ r cos(θ) ≤ √16

0 ≤ r ≤ 4 cos(θ)

Thus, the integral in polar coordinates becomes:

∫(0 to π/2) ∫(0 to 4 cos(θ)) [tex]e^(-r^2[/tex]) r dr dθ

Evaluating the inner integral, we get:

∫(0 to 4 cos(θ)) e^(-r^2) r dr = -1/2 e^(-r^2) |0 to 4 cos(θ)) = 1/2 (1 - e^(-16 cos^2(θ)))

Substituting back into the outer integral, we get:

∫(0 to π/2) 1/2 (1 - e^(-16 cos^2(θ))) dθ

Simplifying, we get:

1/2 ∫(0 to π/2) (1 - e^(-16 cos^2(θ))) dθ

To evaluate this integral, we need to use numerical methods or approximations. Therefore, the final answer will be an approximation.

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The table shows the number of students who signed up for different after school activities. Each student signed up for exactly one activity.
Activity Students
Cooking
9
99
Chess
4
44
Photography
8
88
Robotics
11
1111
Total
32
3232
Match the following ratios to what they describe.
Description
Ratio

Answers

a) The ratio of the photography students to the chess students is 2 : 1

b) The ratio of photography students to all students is 1 : 4

c) The ratio of chess students to all students is 1 : 8

Given data ,

Let the number of cooking students = 9

Let the number of chess students = 4

Let the number of photography students be = 8

Let the number of robotics students = 11

So , the total number of students = 32

a)

The ratio of the photography students to the chess students = 8 / 4

On simplifying the proportion , we get

The ratio of the photography students to the chess students = 2 : 1

b)

The ratio of photography students to all students = 8 / 32

The ratio of photography students to all students = 1 : 4

c)

The ratio of chess students to all students = 4 / 32

The ratio of chess students to all students = 1 : 8

Hence , the proportion is solved

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The complete question is attached below :

The table shows the number of students who signed up for different after school activities. Each student signed up for exactly one activity.

HURRY UP Please answer this question

Answers

Answer:

b=32

Step-by-step explanation:

a^2+b^2=c^2

24^2+b^2=40^2

576+b^2=1600

b^2=1600-576

b^2=1024

b=32

If x = t^3 - t and y = Squareroot 3t + 1, then dy/dx at t = 1 is 3/8 1/8 8/3 8 3/4

Answers

If x = [tex]t^3[/tex] - t and y = [tex]\sqrt{3t + 1}[/tex], then dy/dx is 3/8 at t = 1.

To find dy/dx at t=1, we need to first find dx/dt and dy/dt, and then use the chain rule.
1. Find dx/dt:
x = [tex]t^3[/tex] - t
Differentiate with respect to t:
dx/dt = [tex]3t^2[/tex] - 1
2. Find dy/dt:
y = sqrt(3t + 1)
Differentiate with respect to t:
dy/dt = 1/2 * [tex](3t + 1)^{(-1/2)}[/tex] * 3
dy/dt = 3/(2 * [tex]\sqrt{(3t + 1)}[/tex])
3. Use the chain rule to find dy/dx:
dy/dx = (dy/dt) / (dx/dt)
4. Plug in the values at t = 1:
dx/dt (t=1) = [tex]3(1)^2[/tex] - 1 = 2
dy/dt (t=1) = 3/(2 * [tex]\sqrt{(3(1) + 1)}[/tex]) = 3/4
5. Calculate dy/dx at t = 1:
dy/dx = (3/4) / 2 = 3/8
So, the answer is 3/8.

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